Homotopy theory with marked additive categories Ulrich Bunke∗ Alexander Engely Daniel Kasprowskiz Christoph Wingesx November 9, 2018 Abstract We construct combinatorial model category structures on the categories of (marked) categories and (marked) pre-additive categories, and we characterize (marked) addi- tive categories as fibrant objects in a Bousfield localization of pre-additive categories. These model category structures are used to present the corresponding 1-categories obtained by inverting equivalences. We apply these results to explicitly calculate limits and colimits in these 1-categories. The motivating application is a systematic construction of the equivariant coarse algebraic K-homology with coefficients in an additive category from its non-equivariant version. Contents 1 Introduction2 2 Marked categories5 2.1 Categories of marked categories and marked pre-additive categories . .5 2.2 The model categories preAdd+ and Cat+ ..................8 2.3 (Marked) additive categories as fibrant objects . 23 ∗Fakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, Germany [email protected] yFakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, Germany [email protected] zRheinische Friedrich-Wilhelms-Universit¨atBonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany [email protected] xRheinische Friedrich-Wilhelms-Universit¨atBonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany [email protected] 1 2.4 1-categories of (marked) pre-additive and additive categories . 27 3 Applications 29 3.1 Localization preserves products . 29 3.2 Rings and Modules . 30 3.3 G-coinvariants . 33 3.4 G-invariants . 37 References 43 1 Introduction If C is a category and W is a set of morphisms in C, then one can consider the localization functor −1 `C : C!C1 := C[W ] in 1-categories [Lur14, Def. 1.3.4.1] [Cis, Def. 7.1.2]), where we consider C as an 1- category given by its nerve (which we will omit in the notation). If the relative category (C;W ) extends to a simplicial model category in which all objects are cofibrant, then we have an equivalence of 1-categories coh cf C1 ' N (C ) ; where the right-hand side is the nerve of the simplicial category of cofibrant/fibrant objects of C [Lur14, Def. 1.3.4.15 & Thm. 1.3.4.20]. This explicit description of C1 is sometimes very helpful in order to calculate mapping spaces in C1 or to identify limits or colimits of diagrams in C1. In the present paper we consider the case where C belongs to the list fCat; Cat+; preAdd; preAdd+g where Cat(+) is the category of small (marked) categories (Definition 2.3), and preAdd(+) is the category of small (marked) pre-additive categories (Definitions 2.4 and 2.6), and W are the (marking preserving) morphisms (functors or Ab-enrichment preserving functors, respectively) which admit inverses up to (marked) isomorphisms (Definition 2.15). In order to fix set-theoretic issues we choose three Grothendieck universes U ⊂ V ⊂ W : (1.1) The objects of C are categories in V which are locally U-small, while C itself belongs to W and is locally V-small. We will shortly say that the objects of C are small (as already done above), and correspondingly, that C itself is large. Our first main theorem is: 2 Theorem 1.1. The pair (C;W ) extends to a combinatorial, simplicial model category structure. We refer to Theorem 2.16 for a more precise formulation and recall that the adjective combinatorial means cofibrantly generated as a model category, and locally presentable as a category. In this model category structure all objects of C are cofibrant. The assertion of Theorem 1.1 in the case of Cat and preAdd is well-known or folklore. In the proof, which closely follows the standard line of arguments, we therefore put the emphasis on checking that all arguments work in the marked cases as well. In order to describe the homotopy theory of (marked) additive categories, we show the following. Proposition 1.2. There exists a Bousfield localization LpreAdd(+) of preAdd(+) whose fibrant objects are the marked (additive) categories. We refer to Proposition 2.56 for a more precise statement. Let WAdd(+) denote the weak equivalences in LpreAdd(+). Proposition 1.2 then implies that we have an equivalence of 1-categories Add(+) := preAdd(+)[W −1 ] ' Ncoh(Add(+)) ; (1.2) 1 Add(+) where Add(+) denotes the category of small (marked) additive categories (see Defini- + tions 2.50 and 2.52). For example, this allows us to calculate limits in Add1, which is one of the motivating applications of the present paper (see Example 3.29). Since in general an 1-category modeled by a combintorial model category is presentable, we get the following (see Corollary 2.60). Corollary 1.3. The 1-categories in the list + + + fCat1; Cat1; preAdd1; preAdd1; Add1; Add1g are presentable. Presentability is a very useful property if one wants to show the existence of adjoint functors. For example the inclusion F⊕ : Add1 ! preAdd1 preserves limits (by inspection) and therefore has a left-adjoint, the additive completion functor L⊕ : preAdd1 ! Add1 (see Corollary 2.60). We demonstrate the utility of the model category structures, whose existence is asserted in Theorem 1.1, in a variety of examples. 3 1. In Proposition 2.61, we use relation (1.2) in order to show an equivalence of 1- categories Add1 ' N2(Add(2;1)) ; where the right-hand side is the 2-categorical nerve of the strict two-category of small additive categories. This is used in [BEKW18] to extend K-theory functors from Add to N2(Add(2;1)). 2. In Section 3.1 we verify that the localization functor `C : C!C1 preserves arbitrary products, where C belongs to the list + + + fCat; Cat ; preAdd1; preAdd1; Add1; Add1g ; see Proposition 3.1. 3. In Section 3.2 we consider additive categories of modules over rings. For example, we show in Proposition 3.2 that fg;free L⊕(`preAdd(R)) ' `Add(Mod (R)) ; i.e. that the additive completion of a ring (considered as an object `preAdd(R) in preAdd1) is equivalent to the additive category of its finitely generated and free modules (considered in Add1). We also discuss idempotent completions and its relation with the additive category of finitely generated projective modules along the same lines, see Proposition 3.8. 4. The main result in Section 3.3, see Theorem 3.10, is an explicit formula for the object colim `preAdd(+);BG(A) BG in preAdd(+), where A is a (marked) pre-additive category with trivial action of a group G and `preAdd(+);BG is induced from `preAdd(+) . + + 5. In Section 3.4 we consider C in fpreAdd1; preAdd1; Add1; Add1g. In Theo- rem 3.21, we provide an explicit formula for the object lim `C;BG(A) ; BG where A is an object of C with an action of G. In a parallel paper [Bun16] we consider model categoy structures on (marked) ∗-categories and a similar application to coarse homology theories including equivariant coarse topolog- ical K-homology. Acknowledgements U. Bunke and A. Engel were supported by the SFB 1085 \Higher Invariants" funded by the Deutsche Forschungsgemeinschaft DFG. C. Winges acknowledges support by the Max Planck Society and by Wolfgang L¨uck's ERC Advanced Grant \KL2MG-interactions" (no. 662400). D. Kasprowski and C. Winges are members of the Hausdorff Center for Mathematics at the University of Bonn. 4 2 Marked categories 2.1 Categories of marked categories and marked pre-additive categories In this section we introduce categories of marked categories, marked pre-additive categories and additive categories. We further describe various relations between these categories given by forgetful functors and their adjoints. We finally describe their enrichments in groupoids and simplicial sets. Let C be a category. Definition 2.1. A marking on C is the choice of a wide subgroupoid C+ of the underlying groupoid of C. Example 2.2. In this example, we name the two extreme cases of markings. On the one + hand, we can consider the minimal marking Cmin given by the identity morphisms of C. + On the other hand, we have the maximal marking Cmax given by the underlying groupoid of C. Definition 2.3. A marked category is a pair (C; C+) of a category and a marking. A morphism between marked categories (C; C+) ! (D; D+) is a functor C ! D which + + sends C to D . We let Cat+ denote the category of marked small categories and morphisms between marked categories. We have two functors + + F+ : Cat ! Cat ; (C; C ) 7! C (2.1) and (−)+ : Cat+ ! Groupoids ; (C; C+) 7! C+ : The functor F+ (which forgets the markings) fits into adjunctions + + mi: Cat Cat : F+ ; F+ : Cat Cat: ma ; where the functors mi (mark identities) and ma (mark all isomorphisms) are given (on objects) by + + mi(C) := (C; Cmin) ; ma(C) := (C; Cmax) ; and their definition on morphisms as well as the unit and counit of the adjunctions are the obvious ones. Definition 2.4. A pre-additive category is a category which is enriched over the category of abelian groups. A morphism between pre-additive categories is a functor which is compatible with the enrichment. 5 We let preAdd denote the category of small pre-additive categories and functors which are compatible with the enrichment. The forgetful functor (forgetting the enrichment) is the right-adjoint of an adjunction LinZ : Cat preAdd: FZ (2.2) whose left-adjoint is called the linearization functor. For a pre-additive category A we call FZ(A) the underlying category. Remark 2.5. Let A be a pre-additive category. If A and B are two objects of A such that the product A × B and the coproduct A t B exist, then the canonical morphism A t B ! A × B induced by the maps (idA; 0): A ! A × B and (0; idB): B ! A × B is an isomorphism.